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Implications of an arithmetical symmetry of the commutant for modular invariants. (English) Zbl 1043.81698

Summary: We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU\((3)_k\), we classify the modular invariant partition functions when \(k+3\) is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
11F22 Relationship to Lie algebras and finite simple groups
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81R99 Groups and algebras in quantum theory
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